White Shark Number of Breeders Re-Analysis 2024

Parameter Choices

#use the fixed cohorts
new_cohort_assignment <- TRUE

#use the new fixed genomes
use_fixed_genotypes <- TRUE

#adjust these if you want the relevant chunks to run
rerun_COLONY <- FALSE

#COLONY params for simulations that will actually be used
COLONY_run_length <- 2
COLONY_allele_freq <- NULL # vs "overall" for population averages
COLONY_dropout <- 0 
COLONY_error_rate <- 0.02
COLONY_random_seed <- 1

#see scenarios section for parameter choices
rerun_COLONY_all <- FALSE

rerun_LDNE <- FALSE

#p_crit choices
p_crit_list <- c(0, 0.02, 0.05) 
p_crit_chosen <- 0.05

include_2014 <- TRUE #will still run 2014 estimates, just not included in the analysis





#using old data for cohorts and genotypes, no 2014, pcrit 2014, defaults for COLONY should regenerate the original published results exactly

Overview

This document/repository contains and outlines the re-analysis of the data presented in Davenport et al 2021 Davenport et al. (2021), as re-re-analysed by Andrew Jones.

The authors were made aware of a bug or error in the output-files used in both NeEstimator and Colony to estimate the number of breeders per cohort. This error caused the data to become shuffled. Below, analysis is presented comparing the published data to files made directly from the original raw data. The input files for Nb estimation for NeEstimator and COLONY are re-made using raw data (based on the published loci) and the updated results are presented. The published work further sought to combine estimators of effective size to make reporting and decision making easier - combined estimates of Nb per-cohort. Here using the revised results we also make these re-estimates.

An additional sensitivity analysis is performed on the full dataset to illustrate how the inclusion of different SNPs and individuals can affect final point-estimates of effective size, and highlights the importance of considering confidence intervals (i.e using a 95%CI we expect that 95% of the time the true value of the parameter lies in its interval), as well as various estimators in conservation decisions. S

Background

In Davenport et al., 2021, the effective number of breeders (Nb) was estimated in 4 consecutive cohorts (2010-2013) based on the premise that when a genetic sample contains only individuals from a single age cohort (a group of individuals having the same age-class), then the estimate of effective population size (Ne) from a single-sample estimator in species with overlapping generations corresponds to the effective number of breeders (Nb). For long-lived, iteroparous species such as the white shark, estimates of Nb are generally considered more useful for monitoring as they apply to a single breeding season (rather than needing a sample representative of a generation - an assumption of Ne (see Waples (2022))), and it represents an accessible parameter for monitoring population trends at ecological timescales most relevant to conservation and management needs. Nb is particularly useful in cases where juvenile or YOY samples can be easily collected (i.e. through SMART shark program NSW). Estimating Nb and monitoring its change over time allows timely detection of population trends (decline, restoration, recovery, expansion), even if using as few as four or five consecutive reproductive cycles (Leberg (2005); J. Wang (2005); Luikart et al. (2021)).

Data

The original DArT data for this project can be found in the file /Data/Raw

The results of the reanalysis can be found in /Results.

The filtered list of loci from the published analysis can be found in Data/Raw/Report-DSha18-3402/File_Formats/Genepop_NeEstimator_Input/nswdpi_whiteshark_gp.gen.

The loci names were extracted from this document to ensure the data from the publication are used in the reanalysis here. Readers are referred to the original publication Davenport et al. (2021) for more detailed methods. In summary, a list of loci/SNPs which meet the articles filtering criteria are saved in the ‘.gen’ file. These are used here to re-make the input files for NeEstimator and COLONY from the original raw-data, from which Nb per cohort is re-estimated.The full raw data is also used to re-estimate Nb using a sensitivity analysis of loci/individuals.

The processed Data is found in /Data/processed.

Confirmation of Issue in Original Manuscript

For this section I follow the approach used by Dani as far as was required to confirm the work of Dean and Paul.

Setup data

A bug in an R-package used in the original publication is suspected to have introduced inconsistencies into the output files used in the final analysis.

# read in published strata
strata_info = read.table("Original Project Files/Strata_Sex_Model_COHORT_REGRESSION.tsv", header = T) # used in the published ms 

# read in published genepop file affected by bug - herein refered to as 'orig-genepop'
orig_gp <- adegenet::read.genepop(file = "Original Project Files/File_Formats/Genepop_NeEstimator_Input/nswdpi_whiteshark_gp.gen", ncode = 3L, quiet = TRUE)
orig_gl <- dartR::gi2gl(orig_gp)

## Starting :: 
##  Starting dartR 
##  Starting gi2gl 
## Starting gl.compliance.check 
##   Processing genlight object with SNP data
##   The slot loc.all, which stores allele name for each locus, is empty. 
## Creating a dummy variable (A/C) to insert in this slot. 
##   Checking coding of SNPs
##     SNP data scored NA, 0, 1 or 2 confirmed
##   Checking locus metrics and flags
##   Recalculating locus metrics
##   Checking for monomorphic loci
##     Dataset contains monomorphic loci
##   Checking for loci with all missing data
##     No loci with all missing data detected
##   Checking whether individual names are unique.
##   Checking for individual metrics
##   Warning: Creating a slot for individual metrics
##   Checking for population assignments
##     Population assignments confirmed
##   Spelling of coordinates checked and changed if necessary to 
##             lat/lon
## Completed: gl.compliance.check 
## Completed: :: 
##  Completed: dartR 
##  Completed: gi2gl

# it didnt fix the names, so do it here
orig_gl@loc.names<- sub(".*?-", "", str_replace_all(orig_gl@loc.names,"__","-"))  # change whitelist loci seperators so they are in the fornat needed to match the genlight obj

# 'new_genepop'
# read in orgional data, and use the list of loci to make a new dataset - herein refered to as 'new-genepop'
input_gl = dartR::gl.read.dart("Data/Raw/Report_DSha18-3402_SNP_2_ReLabeled.csv")

## Starting :: 
##  Starting dartR 
##  Starting gl.read.dart 
## Starting utils.read.dart 
##   Topskip not provided.
##  Setting topskip to 6 .
##   Reading in the SNP data
##   Detected 2 row format.
## Number of rows per clone (should be only  2 s): 2 
##  Added the following locus metrics:
## AlleleID AlleleSequence TrimmedSequence Chrom_Shark_Callorhinchus_v1 ChromPos_Shark_Callorhinchus_v1 AlnCnt_Shark_Callorhinchus_v1 AlnEvalue_Shark_Callorhinchus_v1 SNP SnpPosition CallRate OneRatioRef OneRatioSnp FreqHomRef FreqHomSnp FreqHets PICRef PICSnp AvgPIC AvgCountRef AvgCountSnp RepAvg .
## Recognised: 278 individuals and 9841  SNPs in a 2 row format using Data/Raw/Report_DSha18-3402_SNP_2_ReLabeled.csv 
## Completed: utils.read.dart 
## Starting utils.dart2genlight 
## Starting conversion....
## Format is 2 rows.
## Please note conversion of bigger data sets will take some time!
## Once finished, we recommend to save the object using save(object, file="object.rdata")
## Completed: utils.dart2genlight 
##  Data read in. Please check carefully the output above
##   Read depth calculated and added to the locus metrics
##   Minor Allele Frequency (MAF) calculated and added to the locus metrics
##   Recalculating locus metrics provided by DArT (optionally specified)
## Completed: :: 
##  Completed: dartR 
##  Completed: gl.read.dart

# change the GL loci names so they are in a suitable format 
input_gl@loc.names <- sub("^([^-]*-[^-]*).*", "\\1", input_gl@loc.names)
# filter the gl to have the same loci as used in 'orig-gp'
order_loci <- sapply(orig_gl@loc.names, function(x,order){which(order == x)}, order = input_gl@loc.names)
new_gl <- input_gl[,order_loci]
# dartR has a bug where it doesnt fix up other information stored in other
dim(new_gl@other$loc.metrics)

## [1] 3668   24

new_gl@other$loc.metrics <- new_gl@other$loc.metrics[order_loci,]
# match so the files have the same samples, in the same order
order_ind <- sapply(orig_gl@ind.names, function(x,order){which(order == x)}, order = str_replace_all(new_gl@ind.names, "_", "-"))

new_gl <- new_gl[order_ind,]
# check
dim(new_gl@other$loc.metrics)

## [1] 3668   24

# add the population information, also make sure its in the right order
pop = dplyr::left_join(tibble::tibble(TARGET_ID=str_replace_all(new_gl@ind.names, "_", "-")), strata_info, by = "TARGET_ID")
new_gl@pop<- factor(pop$STRATA)
# check 
which(head(str_replace_all(new_gl@ind.names, "_", "-")) ==  head(orig_gl@ind.names))

## [1] 1 2 3 4 5 6

which(head(orig_gl@loc.names) == head(new_gl@loc.names))

## [1] 1 2 3 4 5 6

Quantify differences in missingness between the published and new dataset

# compare the old and the new files for missingness
# print out each genlight to get info on missngness, loci, invidual
orig_gl

##  /// GENLIGHT OBJECT /////////
## 
##  // 241 genotypes,  3,668 binary SNPs, size: 1.2 Mb
##  4506 (0.51 %) missing data
## 
##  // Basic content
##    @gen: list of 241 SNPbin
##    @ploidy: ploidy of each individual  (range: 2-2)
## 
##  // Optional content
##    @ind.names:  241 individual labels
##    @loc.names:  3668 locus labels
##    @loc.all:  3668 alleles
##    @pop: population of each individual (group size range: 1-63)
##    @other: a list containing: loc.metrics  loc.metrics.flags  verbose  history  ind.metrics

# compare the old and the new files for missingness
# print out each genlight to get info on missngness, loci, invidual
new_gl

##  ********************
##  *** DARTR OBJECT ***
##  ********************
## 
##  ** 241 genotypes,  3,668 SNPs , size: 5.7 Mb
## 
##     missing data: 4503 (=0.51 %) scored as NA
## 
##  ** Genetic data
##    @gen: list of 241 SNPbin
##    @ploidy: ploidy of each individual  (range: 2-2)
## 
##  ** Additional data
##    @ind.names:  241 individual labels
##    @loc.names:  3668 locus labels
##    @loc.all:  3668 allele labels
##    @position: integer storing positions of the SNPs [within 69 base sequence]
##    @pop: population of each individual (group size range: 1-63)
##    @other: a list containing: loc.metrics, ind.metrics, loc.metrics.flags, verbose, history 
##     @other$ind.metrics: service, plate_location 
##     @other$loc.metrics: AlleleID, AlleleSequence, TrimmedSequence, Chrom_Shark_Callorhinchus_v1, ChromPos_Shark_Callorhinchus_v1, AlnCnt_Shark_Callorhinchus_v1, AlnEvalue_Shark_Callorhinchus_v1, SNP, SnpPosition, CallRate, OneRatioRef, OneRatioSnp, FreqHomRef, FreqHomSnp, FreqHets, PICRef, PICSnp, AvgPIC, AvgCountRef, AvgCountSnp, RepAvg, clone, uid, rdepth 
##    @other$latlon[g]: no coordinates attached

These results show the number of individuals and the number of loci does not change between the published and the new dataset. However, the number of missing data in the published dataset is 4506, while the re-analysis dataset missingingess is 4503.

Visualise the difference between the published and new dataset

# visualize the data
ogplot = adegenet::glPlot(orig_gl, posi="topleft")

newplot = adegenet::glPlot(new_gl, posi="topleft")

Discussion of error

More details available in Dani’s report and Dean and Paul’s report.

Main points:

• Error definitely present as described in analyis by Dean and Paul.
• Error changes data in a way that would be expected to have some impact estimates of Nb
• Error needs to be corrected

Corrected Ne Estimates

New cohort assignment

The new cohort assignment file “Data/Processed/New_Cohort_Assignment.tsv” is created by running the the following chunks.

The von Bertanlaffy parameters used are from are from O’Connor (2011). Further work could be done exploring the impact of parameter choices, however initial work suggests the estimates are fairly robust to small numbers of individuals being allocated to different cohorts.

meta_data <- readr::read_csv("Data/Raw/Copy of UPDTAED LAT LONGS UQ genetics PhD181018 Paul Butcher NSW DPI.csv") %>%
  #select(MBB_Code, FL, TL) %>%
  mutate(date_column = as.Date(`Date tagged`, format="%d/%m/%y"),
         modified_year = year(date_column))

# 1. relationship between TL and FL 
lm1 <- lm(meta_data$TL ~ meta_data$FL)

Tl_intercept <- lm1$coefficients[1]
Tl_slope <- lm1$coefficients[2]
#
Tl_intercept <- floor(Tl_intercept * 100) / 100
Tl_slope <- floor(Tl_slope * 100) / 100 # eq (1) manuscript , round down to two decimal places

TL <- sapply(meta_data$FL, function(x) {Tl_intercept + x * Tl_slope}) # eq (1) manuscript 

# Estimate age directly using values from O'Connor 2011 and FL
#solve for age
vbgf3 <- function(L, L_infinity, K_value, t_0) {
  result <- t_0 - (1 / K_value) * log(1 - (L / L_infinity))
  return(result)
}
# Estimating age at length
# info:
# Best fitting parmaters for Males
Linf <- 798.94# cm TL
k <- 0.047 # 
L0 <- 140 #cm 
T0 <- -3.8 #years 
Males <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

# Best fitting paramters for Females 
Linf<- 719.02# cm TL
k<- 0.056 # 
L0<- 140 #cm 
T0 <- -3.8 #years 
Females <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

Linf<- 746.66# cm TL
k<- 0.053# 
L0<- 140 #cm 
T0 <- -3.8 #years 
Combined <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

# female 
idx_f<- which(meta_data$Sex == "F")
idx_m<- which(meta_data$Sex == "M")

male_t = sapply(TL[idx_m], function(x) {vbgf3(x, Males[["L_infinity_value"]], Males[["K_value"]], Males[["t_0_value"]])})
female_t = sapply(TL[idx_f], function(x) {vbgf3(x, Females[["L_infinity_value"]], Females[["K_value"]], Females[["t_0_value"]])})

# Cohort is equal to year sampled minus age
male_age = trunc(meta_data$modified_year[idx_m] -  male_t)
female_age = trunc(meta_data$modified_year[idx_f] -  female_t) # including some samples, ie. MBB1348 depends on how you round the age or the year
output = tibble::tibble(STRATA = c(male_age, female_age), MBB_CODE = c(meta_data$MBB_Code[idx_m], meta_data$MBB_Code[idx_f]))
readr::write_tsv(output, "Data/Processed/New_Cohort_Assignment.tsv")

New Input files for NeEstimator and COLONY

Below the input files for NeEstimator and COLONY are remade to make new estimates of Nb using a new file made from the published list of loci (“new_gp”).

dartR is used write out a genepop file, and a custom script based on the radiator package is used to write out a COLONY file. These COLONY files also contain the COLONY parameter settings.

# 1. write out new_gp as a genepop , check that the file has been written out correctly 
# 1a. run this file through NeEstimator

# first its important to make sure that the cohort are correctly assigned 

#some sharks no cohort - why ?no meta data?

if(new_cohort_assignment){
  cohort_assignment_file <- "Data/Processed/New_Cohort_Assignment.tsv"
  
  cohort_assignment = readr::read_tsv(cohort_assignment_file) %>% #New cohort assignment
  separate(MBB_CODE, into = c("string1", "string2"), sep = " ", remove = FALSE) %>%
  unite(MBB_CODE_JOIN, string1, string2, sep = "_") 
  
}else{
  cohort_assignment_file <- "Original Project Files/Strata_Sex_Model_COHORT_REGRESSION.tsv"
  
  cohort_assignment = readr::read_tsv(cohort_assignment_file) %>% #New cohort assignment
  separate(INDIVIDUALS, into = c("string1", "string2"), sep = "-", remove = FALSE) %>%
  unite(MBB_CODE_JOIN, string1, string2, sep = "_") 
}



if(use_fixed_genotypes){
  gl_file <- new_gl
}else{
  gl_file <- orig_gl
}

gl_file$ind.names <- str_replace(gl_file$ind.names, "-", "_")


ind_order = tibble::tibble(MBB_CODE_JOIN = gl_file$ind.names, OLD_STRATA = gl_file@pop) %>%
  left_join(., cohort_assignment, by = "MBB_CODE_JOIN")

gl_file@pop <- as.factor(ind_order$STRATA)
table(gl_file@pop)

## 
## 2005 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 
##    1    3    6   11   29   41   52   64   23    9    2

dartR::gl2genepop(gl_file, outfile = "genotypes_for_Ne.gen", outpath = "Data/Processed/NeEstimator")

## Starting :: 
##  Starting dartR 
##  Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  Data/Processed/NeEstimator/genotypes_for_Ne.gen/
## Completed: :: 
##  Completed: dartR 
##  Completed: gl2genepop

A number of different parameters setttings were used.

For error and drop out rate, the default settings, as were used originally are based on the recommendations of Jinliang Wang (2004). More recently Jinliang Wang (2018) has done further empirical work, and the parameters were also tried as well as some inbetween.

For each of the error and drop out rate options, a version was run with population allele frequencies only based on the cohort (allele.freq = NULL, default setting) and using estimates based on the entire population (allele_freq = “overall”).

The run length parameter was also experimented with.

Run COLONY

#need to read COLONY data in from files

colony_results_files <- list.files("./Results/COLONY/")

col_list <- list()

for (i in seq_len(length(colony_results_files))){
  c_r_f <- colony_results_files[i]
  res <- readLines(file.path("./Results/COLONY/", c_r_f))
  temp <- str_match(c_r_f, "^COLONY_output_(scenario_[a-zA-Z0-9]+)_([0-9]{4}).txt$")
  
  Year <- as.numeric(temp[3])
  Scenario <- temp[2]
  Nb_SA_est <- as.numeric(na.omit(str_match(res, "^Ne\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  Nb_SA_upper <- as.numeric(na.omit(str_match(res, "^CI95\\(U\\)\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  Nb_SA_lower <- as.numeric(na.omit(str_match(res, "^CI95\\(L\\)\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  
  col_list[[i]] <- data.frame(Year, Scenario, Nb_SA_est, Nb_SA_lower, Nb_SA_upper)
}

All_COLONY_data <- bind_rows(col_list) %>% filter(Year >= 2010, Year <= 2014)

write_csv(All_COLONY_data, "./Results/COLONY/ResultsSummary.csv")

Scenario List:

• scenario_00 - Default settings, short run
• scenario_01 - Default settings, population allele frequencies, short run
• scenario_02 - Error rates from Wang 2018, short run
• scenario_03 - Error rates from Wang 2018, population allele frequencies, short run
• scenario_04 - Error rates inbetween 1, short run
• scenario_05 - Error rates inbetween 1, population allele frequencies, short run
• scenario_06 - Error rates inbetween 2, short run
• scenario_07 - Error rates inbetween 2, population allele frequencies, short run
• scenario_08 - Default settings, long run, repeat - Settings from original paper
• scenario_09 - Default settings, long run, repeat
• scenario_10 - Default settings, medium run, repeat
• scenario_11 - Default settings, medium run, repeat

ggplot(All_COLONY_data, aes(x = Year, y = Nb_SA_est, colour = Scenario, group=Scenario)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_SA_lower, ymax = Nb_SA_upper), width = width, position =dodge) 

All_COLONY_data %>% filter(Scenario %in% c("scenario_00", "scenario_01", "scenario_02", "scenario_03","scenario_08", "scenario_final")) %>% ggplot( aes(x = Year, y = Nb_SA_est, colour = Scenario, group=Scenario)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_SA_lower, ymax = Nb_SA_upper), width = width, position =dodge) 

Nb_SA_data <- All_COLONY_data %>% filter(Scenario == "scenario_final") #need to pick one(1) for example scenario_00 is the defaults with 'quick run', scenario_08 is the original settings
write_csv(Nb_SA_data, "./Results/COLONY/ResultsSummary_final.csv")

Estimates slightly lower and more consistent for some years if use population allele frequencies, otherwise basically same. The “short” runs have a degree of inconsistency between runs, but main results will use “medium”. Overall, there is no real reason to deviate from original settings.

Run NeEstimator

Nb_LD_data_all <-  read_csv("./Results/NeEstimator/NeResultsSummary.csv")
Nb_LD_data_all$p_crit <- factor(Nb_LD_data_all$p_crit)

ggplot(Nb_LD_data_all, aes(x = Year, y = Nb_LD_est, colour = p_crit, group=p_crit)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_LD_lower, ymax = Nb_LD_upper), width = width, position =dodge)  + coord_cartesian(ylim = c(0,1000)) + theme_bw()

Nb_LD_data <- Nb_LD_data_all %>% filter(p_crit == p_crit_chosen) #need to pick one(1) 0.05 is the original settings

write_csv(Nb_LD_data, "./Results/NeEstimator/NeResultsSummary_final.csv")

The rare allele bias issue that p_crit fixes is shown here with the different choices for p_crit. Sticking with 0.05 seems like the best option.

Cobmine and Save All Estimates Code

#read in from file so always works even if not a full rerun
Nb_LD_data <- read_csv( "./Results/NeEstimator/NeResultsSummary_final.csv")
Nb_LSA_data <- read_csv( "./Results/COLONY/ResultsSummary_final.csv")

#join and save
data <- left_join(Nb_LD_data, Nb_SA_data, by="Year")
write_csv(data, "./Results/ResultsSummary.csv")

Run Combination Estimates Code

data <- read_csv( "./Results/ResultsSummary.csv")

Ne_to_r <- function(Ne, S){
  r <- (-69*S^2 + sqrt(10000 * (S^4) * (Ne^2) + 4761 * (S^4) - 248400*(S^3)*Ne) + 1800*S*(Ne^2))/( 1800*(S^2)*(Ne^2))  
  return(r)
}

Ne_to_r_star <- function(Ne, S){
  r <- Ne_to_r(Ne, S)
  r <- r - 1/S
  return(r)
}


data$Var_SA_lower <- 0.25 * data$Nb_SA_est^4 * ((1 / data$Nb_SA_lower) - (1 / data$Nb_SA_est))^2
data$Var_SA_upper <- 0.25 * data$Nb_SA_est^4 * ((1 / data$Nb_SA_upper) - (1 / data$Nb_SA_est))^2
data$Var_SA_mean <- ( data$Var_SA_lower + data$Var_SA_upper ) / 2
data$Var_SA_weight <- 1 / data$Var_SA_mean


data$r_LD_lower <- Ne_to_r(data$Nb_LD_lower, data$S) 
data$r_LD_upper <- Ne_to_r(data$Nb_LD_upper, data$S) 
data$r_LD_est <- Ne_to_r(data$Nb_LD_est, data$S) 

data$rstar_LD_lower <- Ne_to_r_star(data$Nb_LD_lower, data$S) 
data$rstar_LD_upper <- Ne_to_r_star(data$Nb_LD_upper, data$S) 


data$Var_LD_lower <- (1/36) * data$r_star^-4 * (data$r_star - data$rstar_LD_lower)^2
data$Var_LD_upper <- (1/36) * data$r_star^-4 * (data$r_star - data$rstar_LD_upper)^2
data$Var_LD_mean <- ( data$Var_LD_lower + data$Var_LD_upper ) / 2
data$Var_LD_weight <- 1 / data$Var_LD_mean

data$weight_sum <- data$Var_LD_weight + data$Var_SA_weight

data$Var_LD_weightadj <- data$Var_LD_weight / data$weight_sum
data$Var_SA_weightadj <- data$Var_SA_weight / data$weight_sum

data$Nb_COM_est <- 1 / ((data$Var_SA_weightadj /data$Nb_SA_est) + (data$Var_LD_weightadj / data$Nb_LD_est))
data$var_COM <- 1 / ((1/data$Var_LD_mean) + (1/data$Var_SA_mean))

#Not CIs
data$Nb_COM_lower <- data$Nb_COM_est - sqrt(data$var_COM) 
data$Nb_COM_upper <- data$Nb_COM_est + sqrt(data$var_COM)

#nice table
data %>% dplyr::select(Year, starts_with("Nb_")) -> data_est_only

if(!include_2014){
  data_est_only <- data_est_only %>% filter(Year != 2014)
}else{
  data_est_only <- data_est_only
}

data_est_only %>% gt() %>% fmt_number(columns = c(5:10), decimals = 1) 

Year	Nb_LD_est	Nb_LD_lower	Nb_LD_upper	Nb_SA_est	Nb_SA_lower	Nb_SA_upper	Nb_COM_est	Nb_COM_lower	Nb_COM_upper
2010	61.2	31.9	257.4	96.0	58.0	194.0	74.7	54.9	94.5
2011	205.7	109.8	1023.9	234.0	144.0	492.0	222.5	169.2	275.9
2012	214.8	134.0	489.2	196.0	133.0	319.0	199.1	160.8	237.5
2013	171.9	117.3	301.7	152.0	109.0	221.0	154.2	128.9	179.5
2014	103.3	48.8	10445.4	127.0	71.0	358.0	114.3	81.4	147.2

Results

Summary Plots

Plot of Nb estimates with 95% confidence intervals

The combined estimates are also plotted here. The error bars in this case represent +- SD, not a confidence interval.

Plot of Combined Nb estimates +- SD

Stability

I take stability to mean that there are no significant differences between the Nb estimates for any years, this corresponds to the null hypothesis that Nb_2010 = Nb_2011 = Nb_2012 = Nb_2013.

LD (NeEstimator)

The situation is very difficult to evaluate graphically. No hard conclusions can be drawn from examining the individual CIs alone, however there is no obvious trends and the band of values that are in all confidence intervals is relatively wide.

LD estimates with 95% CIs and values which fall in all CIs marked as a grey rectangle

In order to examine this formally, we will need to construct pairwise tests. This needs to be done with \(r^{*}\) as LDNe confidence intervals do not have normal CIs. The confidence intervals for \(r^{*}\) in LDNe are approximately normal. This method of comparison is statistically consistent with manner used in the construction of the CIs [cite: Jones, Ovenden and Wang (2016)].

A slightly more accurate result could be obtained by pulling the results directly from the LDNe code, and making minor adjustments to account for the fact this is only approximately normal, however I do not believe this would make a difference in this case.

The following table summarises the pairwise comparisons of the Nb estimates in terms of Nb. The first two columns identify which years are being compared, the third and forth columns state the relevant \(r^{*}\) values.

Their difference and the estimate for the standard deviation of the estimate follow. The combined standard deviation is found by taking the square root of the sum the of the two variances. This assumes both estimates are independent. While it is likely these estimates are correlated with each other, we have no way of accurately estimating their covariance so I believe it is best to assume independence. Holm method on the basis that we are making 6 pairwise comparisons and we wish to keep our family-wise error rate below 0.05.

data$rstar_LD_lower <- Ne_to_r_star(data$Nb_LD_lower, data$S) 
data$rstar_LD_upper <- Ne_to_r_star(data$Nb_LD_upper, data$S) 
data$rstar_LD_est <- Ne_to_r_star(data$Nb_LD_est, data$S) 
data$rstar_sd_mean <- (data$rstar_LD_lower - data$rstar_LD_upper) / (2*qnorm(0.9750))

results_frame <- data.frame("Year1" = NA, "Year2"= NA, "r1"= NA, "r2"= NA, "diff"= NA, "combined sd"= NA, "z-score"= NA, "p-value"= NA) %>% na.omit()

for(i in 1:(nrow(data)-1)){
  for(j in (i+1):nrow(data)){
    temp <- c(
      data$Year[i],
      data$Year[j],
      data$rstar_LD_est[i],
      data$rstar_LD_est[j],
      data$rstar_LD_est[i]-data$rstar_LD_est[j],
      sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2), 
      (data$rstar_LD_est[i]-data$rstar_LD_est[j]) / sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2),
      1-pnorm(abs(data$rstar_LD_est[i]-data$rstar_LD_est[j]) / sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2))
      
    )
    
    results_frame[nrow(results_frame) + 1,] <- temp
  
  }
}

results_frame$p.value.adj <- p.adjust(results_frame$p.value, "holm")
results_frame %>% gt() %>% opt_stylize() %>% fmt_number(columns = c(7:9), decimals = 3) %>% fmt_number(columns = c(3:6), n_sigfig =  3)

Year1	Year2	r1	r2	diff	combined.sd	z.score	p.value	p.value.adj
2010	2011	0.000891	0.000269	0.000622	0.000391	1.592	0.056	0.501
2010	2012	0.000891	0.000258	0.000634	0.000382	1.661	0.048	0.484
2010	2013	0.000891	0.000322	0.000570	0.000381	1.495	0.067	0.539
2010	2014	0.000891	0.000531	0.000360	0.000468	0.768	0.221	1.000
2011	2012	0.000269	0.000258	0.0000113	0.000137	0.082	0.467	1.000
2011	2013	0.000269	0.000322	−0.0000527	0.000135	−0.389	0.349	1.000
2011	2014	0.000269	0.000531	−0.000263	0.000304	−0.864	0.194	1.000
2012	2013	0.000258	0.000322	−0.0000640	0.000106	−0.607	0.272	1.000
2012	2014	0.000258	0.000531	−0.000274	0.000292	−0.938	0.174	1.000
2013	2014	0.000322	0.000531	−0.000210	0.000291	−0.721	0.235	1.000

SA (COLONY)

With the updated results, the intervals for the SA method are no longer mutually disjoint. Analysis is much the same as for LD now.

The analysis proceeds in a similar manner as for \(r^{*}\).

data$Nb_SA_est_inv <- 1 / (2*data$Nb_SA_est)

data$vstar_lower <- ((1/data$Nb_SA_lower - 1/data$Nb_SA_est) / (2*qnorm(0.9750)))^2
data$vstar_upper <- ((1/data$Nb_SA_est - 1/data$Nb_SA_upper) / (2*qnorm(0.9750)))^2

data$vstar_mean <- (data$vstar_upper+data$vstar_lower)/2

results_frame <- data.frame("Year1" = NA, "Year2"= NA, "2Nb^-1 1"= NA, "2Nb^-1 2"= NA, "diff"= NA, "combined sd"= NA, "z-score"= NA, "p-value"= NA) %>% na.omit()

for(i in 1:(nrow(data)-1)){
  for(j in (i+1):nrow(data)){
    temp <- c(
      data$Year[i],
      data$Year[j],
      data$Nb_SA_est_inv[i],
      data$Nb_SA_est_inv[j],
      data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j],
      sqrt(data$vstar_mean[i] + data$vstar_mean[j]), 
      (data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j]) / sqrt(data$vstar_mean[i] + data$vstar_mean[j]),
      1-pnorm(abs(data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j]) / sqrt(data$vstar_mean[i] + data$vstar_mean[j]))
      
    )
    
    results_frame[nrow(results_frame) + 1,] <- temp
  
  }
}

results_frame$p.value.adj <- p.adjust(results_frame$p.value, "holm")

names(results_frame)[3:4] <-c('(2Ne)^-1 1', '(2Ne)^-1 2') 
results_frame %>% gt() %>% opt_stylize() %>% fmt_number(columns = c(6:9), decimals = 3) %>% fmt_number(columns = c(3:5), n_sigfig =  3)

Year1	Year2	(2Ne)^-1 1	(2Ne)^-1 2	diff	combined.sd	z.score	p.value	p.value.adj
2010	2011	0.00521	0.00214	0.00307	0.002	1.832	0.034	0.335
2010	2012	0.00521	0.00255	0.00266	0.002	1.608	0.054	0.486
2010	2013	0.00521	0.00329	0.00192	0.002	1.152	0.125	0.872
2010	2014	0.00521	0.00394	0.00127	0.002	0.599	0.275	0.920
2011	2012	0.00214	0.00255	−0.000414	0.001	−0.491	0.312	0.920
2011	2013	0.00214	0.00329	−0.00115	0.001	−1.329	0.092	0.735
2011	2014	0.00214	0.00394	−0.00180	0.002	−1.141	0.127	0.872
2012	2013	0.00255	0.00329	−0.000738	0.001	−0.901	0.184	0.920
2012	2014	0.00255	0.00394	−0.00139	0.002	−0.893	0.186	0.920
2013	2014	0.00329	0.00394	−0.000648	0.002	−0.414	0.340	0.920

I have a number of reservations about this analysis for the COLONY estimates, but I have included this analysis as I believe it is important.

These issues are:

• I am much less familiar with the CIs for COLONY than for LDNe
• The accuracy of extracting the variance from the CI bounds was not very high, at the time of the original paper I put this down to the fact it rounds to whole numbers, but I’m not longer sure

Trends

I have also tested for (linear) trends. With only five points the power to detect a trend is very low, but I think the exercise is worthwhile.

LD (NeEstimator)

ols <- lm(Nb_LD_est ~ Year, data=data_est_only)

lmfit <- broom::augment(ols, se_fit = TRUE, interval='confidence')

Using the method of Luikart et al 2020 (i.e. ordinary least squares regression on the Nb_LD estimates) there is no significant trend (p-value = 0.848). We can also see this graphically as the CI for the trendline (grey shaded area) includes the possibility of a flat line (i.e. slope=0, no trend).

gls3 <- nlme::gls(Nb_LD_est ~ Year, correlation=corCAR1(0.5, form = ~ 1, fixed=TRUE),  method='REML', verbose=TRUE, data=data)

# Design matrix for our observations ,, 
xmat <- model.matrix(~ Year, data=data)

# Regression coefficients
betahat <- coef(gls3)

glsfit <- broom.mixed::augment(gls3)

Sigmahat <- vcov(gls3)
#https://fw8051statistics4ecologists.netlify.app/gls.html
# var/cov(beta0 + beta1*X)
varcovhat<-xmat%*%Sigmahat%*%t(xmat)
SEline <- sqrt(diag(varcovhat))

glsfit$.upper <- glsfit$.fitted + qnorm(0.975) * SEline
glsfit$.lower <- glsfit$.fitted - qnorm(0.975) * SEline

Nor is there a trend when using a GLS regression that incorporates an estimate of autocorrelation (p-value = 0.808). This model only accounts for correlation. It doesn’t resolve any of the other issues such as asymetric CIs, or incorporate other known information such as the variances of each estimate. However these tests for a trend are likely suitable here as 4 points are very unlikely to show evidence of a trend regardless. I do not believe anyone has published on this as of 2024, let alone at the time of the original publication.

SA (COLONY)

The same analysis can be repeated with the SA estimates.

ols <- lm(Nb_SA_est ~ Year, data=data_est_only)

lmfit <- broom::augment(ols, se_fit = TRUE, interval='confidence')

Using the method of Luikart et al 2020 (i.e. ordinary least squares regression on the Nb_LD estimates) there is no significant trend (p-value = 0.927).

gls3 <- nlme::gls(Nb_SA_est ~ Year, correlation=corCAR1(0.5, form = ~ 1, fixed=TRUE),  method='REML', verbose=TRUE, data=data)

# Design matrix for our observations ,, 
xmat <- model.matrix(~ Year, data=data)

# Regression coefficients
betahat <- coef(gls3)

glsfit <- broom.mixed::augment(gls3)

Sigmahat <- vcov(gls3)
#https://fw8051statistics4ecologists.netlify.app/gls.html
# var/cov(beta0 + beta1*X)
varcovhat<-xmat%*%Sigmahat%*%t(xmat)
SEline <- sqrt(diag(varcovhat))

glsfit$.upper <- glsfit$.fitted + qnorm(0.975) * SEline
glsfit$.lower <- glsfit$.fitted - qnorm(0.975) * SEline

Nor is there a trend when using a GLS regression that incorporates an estimate of autocorrelation (p-value = 0.896).

Discussion

Plot of Nb estimates with 95% confidence intervals

R Settings

R.version

##                _                           
## platform       aarch64-apple-darwin20      
## arch           aarch64                     
## os             darwin20                    
## system         aarch64, darwin20           
## status                                     
## major          4                           
## minor          3.3                         
## year           2024                        
## month          02                          
## day            29                          
## svn rev        86002                       
## language       R                           
## version.string R version 4.3.3 (2024-02-29)
## nickname       Angel Food Cake

References

Davenport, Danielle, Paul Butcher, Sara Andreotti, Conrad Matthee, Andrew Jones, and Jennifer Ovenden. 2021. “Effective Number of White Shark (Carcharodon Carcharias, Linnaeus) Breeders Is Stable over Four Successive Years in the Population Adjacent to Eastern Australia and New Zealand.” Ecology and Evolution 11 (1): 186–98.

Leberg, Paul. 2005. “Genetic Approaches for Estimating the Effective Size of Populations.” The Journal of Wildlife Management 69 (4): 1385–99.

Luikart, Gordon, Tiago Antao, Brian K. Hand, Clint C. Muhlfeld, Matthew C. Boyer, Ted Cosart, Brian Trethewey, Robert Al-Chockhachy, and Robin S. Waples. 2021. “Detecting Population Declines via Monitoring the Effective Number of Breeders (Nb).” Molecular Ecology Resources 21 (2): 379–93. https://doi.org/10.1111/1755-0998.13251.

O’Connor, J. 2011. “Age, Growth and Movement Signatures of the White Shark (Carcharodon Carcharias) in Southern Australia.” The University of Technology Sydney.

Wang, J. 2005. “Estimation of Effective Population Sizes from Data on Genetic Markers.” Philosophical Transactions of the Royal Society B: Biological Sciences 360 (1459): 1395–1409. https://doi.org/10.1098/rstb.2005.1682.

Wang, Jinliang. 2004. “Sibship Reconstruction from Genetic Data with Typing Errors.” Genetics 166 (4): 1963–79.

———. 2018. “Estimating Genotyping Errors from Genotype and Reconstructed Pedigree Data.” Methods in Ecology and Evolution 9 (1): 109–20.

Waples, Robin S. 2022. “What Is N e, Anyway?” Edited by William Murphy. Journal of Heredity 113 (4): 371–79. https://doi.org/10.1093/jhered/esac023.